Best Known (173, 218, s)-Nets in Base 3
(173, 218, 688)-Net over F3 — Constructive and digital
Digital (173, 218, 688)-net over F3, using
- t-expansion [i] based on digital (172, 218, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (172, 220, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 55, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 55, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (172, 220, 688)-net over F3, using
(173, 218, 2119)-Net over F3 — Digital
Digital (173, 218, 2119)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3218, 2119, F3, 45) (dual of [2119, 1901, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3218, 2216, F3, 45) (dual of [2216, 1998, 46]-code), using
- construction XX applied to Ce(45) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(45) ⊂ Ce(40) ⊂ Ce(39) [i] based on
- discarding factors / shortening the dual code based on linear OA(3218, 2216, F3, 45) (dual of [2216, 1998, 46]-code), using
(173, 218, 230108)-Net in Base 3 — Upper bound on s
There is no (173, 218, 230109)-net in base 3, because
- 1 times m-reduction [i] would yield (173, 217, 230109)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 34 302459 510813 088539 578685 692689 225547 081740 438452 508136 250552 850775 783009 076822 989236 896573 819819 888369 > 3217 [i]